Let $F$ be a field and $\zeta \in F$ be a primitive $n$-th root of unity. Also, let $E/F$ be a finite Galois extension with group $G$.
Now I would like to understand the map $f: F^\times \cap E^{\times n} \to H^1(G, \mu_n$ which occurs in Milne's Fields and Galois Theory (page 72).
It was not explicitly mentioned how it is defined but I think it goes as follows: For an element $z \in F^\times \cap E^{\times n}$ there is a $c \in E$ such that $c^n = z$. Now $f(z)$ is defined to be the class of $g: G \to \mu_n$, $\sigma \mapsto \sigma(c)/c$ in $H^1(G,\mu_n)$.
I know that the value is independent of the choice of $c$ and that $\sigma(c)/c \in \mu_n$. However, I am not so sure about the fact that it is indeed a crossed homomorphism. More precisely, I cannot show the equality of $$\frac{\sigma \tau (c)}{c} = \frac{\sigma(c)}{c} \cdot \frac{\tau(c)}{c} $$ for $\sigma,\tau \in G$.
Question Is the map I described the right one (resp. why is $g$ a crossed homomorphism)?